We travelled last year to WDW with our 4 yr old and 18 mos old. All we had for both girls was their birth certificates. I also carry their health cards just in case we needed them b/w leaving from our city and heading to the airport. They only want to see their birth certificates. You shouldn't have any problems at all. They know that little ones don't have other forms of id (as a health card isn't legal id).
Have fun ... our little one was fun last year until she came down with the norwalk bug while were there. She ended up missing the actual day at WDW, but did make it to the Cinderella dinner for my oldest dd's birthday. That was the highlight of the trip for her. The rest was vomitting and "the other" ... lots of fun for all of us

I am hoping she stays healthy for the next trip in March as she is two for two (trip one at 8 mos she ended up with RSV). Lets hope the third time is lucky for her!!
Have a great trip!!
Kerri